Pricing and Demand

University of San Francisco

Matt Meister

Pricing and Demand for a Single Product

Overview

Marketing-mix modeling & demand curves
Demand models
- Linear model
- Multiplicative/log-linear demand model
Price elasticity
Estimation and prediction with the log-linear demand model
Profitability analysis
Extension to multiple products

Marketing-mix modeling (MMM)

Modern approach to manage:
- Prices
- Promotion planning
- Advertising measurement
- Channels of distribution
- Competitive threats
Objectives and approach
- Use data & statistical models to estimate how the marketing mix affects demand
- Understand effect of competitor’s actions on own-demand
- Fine-tune marketing to achieve an objective, in particular increase profits and the value of a product

Demand curves (functions)

A demand curve or demand function relates the quantity of a good sold to various elements of the marketing mix and other variables influencing consumer behavior
$Q = \beta_0 + \beta_1Price + \beta_2Promotion + \beta_3Season + etc$
Both own and the competitor’s marketing actions will typically affect demand

Marketing-mix modeling and demand estimation

Goal
- Infer the relationship between unit sales and prices, promotions, and other variables
To use a statistical tool such as regression analysis we need a mathematical formulation of demand
- This formulation should be flexible and have a good chance of fitting the demand relationship present in common data sets
In this lecture we focus on prices

Demand models

The linear demand model

Formula: $Q = \beta_0 + \beta_1P$

$Q$ is unit sales
$P$ is price
$\beta_0$ is
- $Q$ when $P = 0$
$\beta_1$ is
- Change in $Q$ when $P$ increases by $1
What is the formula in this example?
- $Q = 40 - 5 \times P$

The linear demand model

Formula: $Q = \beta_0 + \beta_1P$

Effect of a given price change is the same for all price points
- Is this a desirable property?

The multiplicative demand model

Formula: $Q = A\times P^{-\eta}$

Effect of a given price change is larger at low than at high price points
- Consistent with niche and mainstream segments of consumers

Logarithms and the exponential function

The log() (logarithm) and exp() (exponential function) will make it easier to manipulate and work with the multiplicative demand model
- Particularly useful: the logarithm transforms multiplication into addition
Notation
- log(x) is always the natural logarithm
- Both $e^x$ and exp(x_ mean the same thing—the exponential function applied to $x$
- Multiplicative demand model = log-linear demand model

Logarithms and the exponential function

We can use logarithms to transform the multiplicative demand model:
- $Q = A\times P^{-\eta}$
- $log(Q) = log(A) + log(P^{-\eta})$
- $log(Q) = log(A) -\eta\times log(P)$
- $log(Q) = \beta_0 +\beta_1\times log(P)$

In R

$log(Q) = \beta_0 +\beta_1\times log(P)$
lm( log(Q) ~ log(P))

We will therefore also refer to the multiplicative demand model as the log-linear demand model—both represent the same demand relationship
Why?
- This model has the form of a linear regression
- We can use lm() to estimate the model, using logged DV/IVs
- $\beta_0$ is the log-transformed intercept for log sales
- $\beta_1$ is the log-transformed effect of price on log sales

Effect of parameters

Parameter $A$ is the intercept

An increase $A$ increases the demand level at any given price

$A$ is higher in the red example.

Effect of parameters

Parameter $\eta$ is like the slope

An increase $\eta$ makes the demand curve steeper—more responsive to price

Price Elasticity

The (own) price elasticity of demand is defined as the percentage change in the quantity demanded relative to a given percentage change in price:
- $price\space elasticity = \frac{\%\triangle Q}{\%\triangle P} = \frac{\frac{Q_1 - Q_0}{Q_0}}{\frac{P_1-P_0}{P_0}}$
- Useful because it does not depend on the level of unit sales or prices
- Typically negative (why?)
Interpretation
- Example: Price elasticity is -2.8
- A 1% increase in price is associated with a 2.8% decrease in unit sales
Note: The price elasticity can be calculated for any demand curve, not just the multiplicative demand model

Categorization of price elasticities

Inelastic demand: $-1 < price\space elasticity < 0$
Elastic demand: $price\space elasticity < -1$

After a price cut:
- Revenue increases for elastic demand
  - Because Q increases a LOT
- Revenue decreases for inelastic demand
  - Because Q does not increase enough

Empirical generalizations about (own) price elasticity

Survey of 367 price elasticity estimates
- (Dolan and Simon, Power Pricing, New York: The Free Press, 1996)
- Average = -1.76
- vs. average advertising expenditure elasticity of +0.22
- Higher for industrial products (2—100)
- Lower(ish) for consumer goods (1.5—5)
- Lowest for luxury (.7—1.5)
Elasticity estimates far outside these ranges may indicate a problem with demand model (e.g. omitted variables)

Price elasticity in the multiplicative demand model

$Q = A\times P^{-\eta}$

$\eta$ has simple and straightforward interpretation—it is the absolute value of the price elasticity of demand
The property that one parameter directly measures the price elasticity is very special and not true for other demand models
- Example: linear demand model
- Elasticity = $-\beta_1 \frac{P}{Q}$

Price elasticities and the relationship between price and sales changes

For any demand model (not just the multiplicative demand model) price and sales changes are related as follows:
$\%\triangle Q = price \space elasticity \times \%\triangle P$
- This relationship is only an approximation
- The approximation is better for small than for large price changes
- Approximations typically reasonable up to 5%-10% price changes
For our purposes and exact predictions, we will use the predict() function in R
- Predict demand at prices $P_0$ and $P_1$, and compute % change

Workshop

Profitability of slot machines in a casino

Should the casino increase or decrease slot machine “prices”?

casino <- read.csv("casino.csv")

head(casino)

  slot_number month days_active hold denomination location  coin_in
1           1     1          28 9.84         0.25        J 69090.45
2           1     2          31 9.84         0.25        J 73333.34
3           1     3          27 8.52         0.25        J 63591.55
4           1     4          31 8.52         0.25        J 84263.49
5           1     5          30 9.84         0.25        J 92990.34
6           1     6          31 9.84         0.25        J 67822.66

summary(casino)

  slot_number       month         days_active        hold      
 Min.   :   1   Min.   : 1.000   Min.   : 1.0   Min.   : 4.25  
 1st Qu.: 701   1st Qu.: 5.000   1st Qu.:30.0   1st Qu.: 7.53  
 Median :1217   Median : 8.000   Median :31.0   Median : 9.99  
 Mean   :1189   Mean   : 8.526   Mean   :29.9   Mean   :10.15  
 3rd Qu.:1743   3rd Qu.:13.000   3rd Qu.:31.0   3rd Qu.:12.43  
 Max.   :2401   Max.   :16.000   Max.   :31.0   Max.   :16.75  
  denomination       location            coin_in       
 Min.   :  0.050   Length:30439       Min.   :    303  
 1st Qu.:  0.250   Class :character   1st Qu.:  47062  
 Median :  0.250   Mode  :character   Median :  77301  
 Mean   :  0.956                      Mean   : 117583  
 3rd Qu.:  1.000                      3rd Qu.: 137849  
 Max.   :100.000                      Max.   :5111112

Some useful observations from the data:

2,401 slot machine observed over a range of 16 months
The average slot machine has been active for 29.9 days in a given month. As low as 1 day a month and as high as 31 days.
The mean hold is 10.15 with a standard deviation of 2.63, and it varies from 4.25 to 16.75
There are 20 locations.
The average revenue per month for a slot machine is $117,582.57. As low as $303.03, as high as $5,111,112.00.

What is the total revenue of the casino chain over the 16 month period?

sum(casino$coin_in * casino$hold/100)

[1] 314009232

Demand Model

What is price in this data?
- There are no explicit prices. The hold of the casino can be considered to be like a price.
  - The hold is the (long-run) average proportion of the bets that the casino keeps.

Profit

If:
- $Q$ is the volume of play (referred to as coin_ins by the industry)
- $h$ is the hold and $c$ is the marginal cost
- Profits of the casino are:
  - $pft = Q_h − Q_c$
- But $c$ of a casino is very close to zero
  - $pft = Q_h$
- Without any marginal costs, we can decide to raise or lower prices from the price elasticity estimate alone

Demand Model

There is variation in the number of days for which the machine is active.
- coin-ins will naturally be higher for a machine that is active for more days.
  - Using coin_in itself can give misleading results.
- Create a new variable Qdaily, which is:

the ratio of coin_ins and days_active

casino$Qdaily <- casino$coin_in/casino$days_active

Model building

Will the data be useful for modeling slot machine demand?
- We need variation in holds in order to run a regression analysis for this purpose
  - We always need variation in the IV and DV!
- How can we check this?

Plot a histogram

hist(casino$hold)

Model building

Will the data be useful for modeling slot machine demand?
- We need variation in holds in order to run a regression analysis for this purpose
  - We always need variation in the IV and DV!
- How can we check this?

Scatter plot of Qdaily vs. hold

ggplot(data = casino, aes( x = hold, y = Qdaily)) +
         geom_point() +
         theme_classic()

What is the relationship?

Is it linear or is it more “curved”?

There seems to be a downward sloping effect of holds on coin_ins
- This is good news!
The relationship does not appear to be linear
- Very small hold: coin_ins high
A multiplicative/log-linear demand model is likely to be a better fit to these data.

Simple demand model specification

Now we can use Qdaily as our demand variable and our demand model can be written as:

$log(Qdaily) = \beta_0 + \beta_1 \times log(hold) + e$

in R:

lm1 <- lm(log(Qdaily) ~ log(hold), data = casino)

Simple demand model specification

summary(lm1)


Call:
lm(formula = log(Qdaily) ~ log(hold), data = casino)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.6346 -0.4141  0.0156  0.4780  3.2330 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 10.96853    0.03555  308.52   <2e-16 ***
log(hold)   -1.34687    0.01547  -87.05   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7674 on 30437 degrees of freedom
Multiple R-squared:  0.1993,    Adjusted R-squared:  0.1993 
F-statistic:  7578 on 1 and 30437 DF,  p-value: < 2.2e-16

Interpret the coefficient for hold – What does this imply for raising or lowering hold percentages?
- This elasticity estimate suggests that the demand is elastic
- The casino should decrease holds to increase revenues/profits
- It suggests that consumers are aware of and very responsive to holds.
What concerns might you have for the validity of this?
- If we have omitted variables, then we can bias our estimates of price elasticities

Prediction tasks

Let’s predict Qdaily when hold <- 2

logQ_predict <- predict(lm1, newdata = data.frame( hold = 2))

How do we take this out of log() form?

Prediction tasks

How do we take this out of log() form?

It would be nice if this worked:

exp(logQ_predict)

       1 
22810.08

But it doesn’t. Why?

In lm(), we assume that the average error is 0
- Not the sum of squared error
When we log-transform and exponentiate, it’s actually $e^{\frac{\sigma^2}{2}}$
- Where $\sigma^2$ is the standard error of the log-linear regression model

Prediction tasks

How do we take this out of log() form?

When we log-transform and exponentiate, avg error = $e^{\frac{\sigma^2}{2}}$
This means that $\frac{\sigma^2}{2}$ is included in our prediction:
- $Q = e^{a-\eta log(P)+\frac{\sigma^2}{2}}$
- If we just type exp( predict(...)), we get $Q = e^{a-\eta log(P)}$
- And underestimate demand
- The fix is somewhat easy

Prediction tasks

How do we take this out of log() form?

$Q = e^{a-\eta log(P)+\frac{\sigma^2}{2}}$

exp(logQ_predict + sigma(lm1) ^ 2 / 2 )

       1 
30619.65

sigma(lm1) is the standard error of lm1 ($\sigma$)

Prediction tasks

Confusing… I know. Just remember:

# Same as always
logQ_predict <- predict(lm1, newdata = data.frame( hold = 2))

# Take it out of log
Q_predict <- exp(logQ_predict + sigma(lm1) ^ 2 / 2 )

What is our prediction?

Q_predict

       1 
30619.65

Controlling for slot denominations

We have information on slot denominations in the data.

Should we include that information in our analysis?
- Do denominations affect the volume of coin-ins?
  - Likely. Consumers who use high-denomination slot machines are likely systematically different.
- Are slot denominations potentially correlated with holds?
  - How could we answer?

Controlling for slot denominations

Generate a scatter plot of demonination vs. hold:

Use factor(denomination)

ggplot(data = casino, aes( x = factor(denomination), y = hold)) +
         geom_point() +
         theme_classic()

Controlling for slot denominations

lm2 <- lm(log(Qdaily) ~ log(hold) + factor(denomination), data = casino)
summary(lm2)


Call:
lm(formula = log(Qdaily) ~ log(hold) + factor(denomination), 
    data = casino)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.0291 -0.3731  0.0387  0.4546  2.1024 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)               9.96298    0.05725 174.039  < 2e-16 ***
log(hold)                -0.66380    0.02121 -31.292  < 2e-16 ***
factor(denomination)0.25 -0.73884    0.02428 -30.429  < 2e-16 ***
factor(denomination)0.5  -0.67726    0.02819 -24.026  < 2e-16 ***
factor(denomination)1    -0.35017    0.02635 -13.289  < 2e-16 ***
factor(denomination)2    -0.22966    0.04725  -4.860 1.18e-06 ***
factor(denomination)5     0.33564    0.03555   9.442  < 2e-16 ***
factor(denomination)10    0.58851    0.05921   9.939  < 2e-16 ***
factor(denomination)25    0.71686    0.07777   9.218  < 2e-16 ***
factor(denomination)100   1.46711    0.11262  13.026  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7286 on 30429 degrees of freedom
Multiple R-squared:  0.2784,    Adjusted R-squared:  0.2781 
F-statistic:  1304 on 9 and 30429 DF,  p-value: < 2.2e-16

Interpret the coefficient for hold
- The coefficient estimate imples a 1% increase in price corresponds to a 0.66% decrease in demand. – What does this estimate imply for the casino?
- Thus, the casino could gain in profitability by raising rather than lowering holds – Which model (lm1 or lm2) do you have greater confidence in? Why?
- In light of lm2 using stronger controls, we will tend to have higher confidence in model lm2

If you want more practice

There is also location (location) and time (month) information.

Practice with adding those if you want more.

Price optimization

With inelastic demand, predicted profits increase forever as prices increase.
- This emphasizes the role of price constraints — Statistical reliability of model decreases when proposed prices are very different from prices in the data.
- In the case of inelastic demand, “optimal” price will generally be the maximum price considered under the constraints.
- In practice, managerial reaction should be to raise prices by a “reasonably large” amount (e.g. ~10-15%)
- After demand data is collected using the new prices, the demand model should be re-estimated.
- This process should be repeated until demand estimates are elastic, at which point truly optimal prices may be calculated.

Price optimization

Although we have inelastic demand in this case, we can still demonstrate the graphical approach to determining optimal prices
To do this, we will:
- Create an R function to evaluate casino profits, given predictors and an lm() model
- Evaluate profits over a range of holds (prices), and plot the results.

Price optimization

Create a function called casino_pft()

Inputs: – hold - hold percentage – month - month of observation – denomination - denomination of slot machine – location - location of slot machine – lm_result - log-linear model estimation results (lm output)
Output: – Q - predicted demand (daily coin-ins) – profit - predicted profit

Price optimization

Create a function called casino_pft()

casino_pft <- function(hold, month, denomination, location, lm_result) {
  # DF that we will use to make predictions
  pred_DF = data.frame( month = month, 
                        hold = hold,
                        denomination = denomination,
                        location = location)
  
  logQ = predict(lm_result, newdata=pred_DF) #Predict log value
  Q = exp(logQ+sigma(lm_result)^2/2) #Fix it!
  profit = Q*hold/100
  return(list(Q = Q, profit = profit))
}

Price optimization

Use your casino_pft() to evaluate casino profits for:

hold: hold percentages ranging from 10 to 20 (percent)
month = 1
denomination = 1
location = “B”
lm_result = lm3

Where lm3:

lm3 <- lm(log(Qdaily) ~ log(hold) + month + factor(denomination) + factor(location), data = casino)

Price optimization

Create data.frame to hold results:

results <- data.frame(
  holds = 10:20,
  pft = rep(0,length(10:20))
)

Calculate and plot!

for (i in 1:nrow(results)) {
  results$pft[i] <- casino_pft(hold = results$hold[i],
                               month = 1,
                               denomination = 1,
                               location = "B",
                               lm_result = lm3)$profit
}

ggplot(data = results, 
       aes( x = holds, y = pft)) +
         geom_point() +
         theme_classic()

Discussion

What does the plot imply for the optimal pricing for the specified slot machine?
- The plot demonstrates that “optimal” profits occur at the maximum hold evaluated (20). In the case of elastic demand, such methods can be used to find optimal profits that do not occur at constraint boundaries.

Pricing and Demand for Multiple Products

Recall from last class: The multiplicative demand model

$log(Q_i) = \beta_0 + \beta_1\times log(P_i)$

This is often too simple
- Does not account for competing products
But we can generalize it to allow for the effect of competing product prices

Pricing and Demand for Multiple Products

Single product:

$log(Q_i) = \beta_0 + \beta_1\times log(P_i)$

Multiple products:

$log(Q_i) = \beta_{i0} + \beta_{i1}\times log(P_{i}) + \beta_{i2}\times log(P_{2})$
- $log(Q_1) = \beta_{10} + \beta_{11}\times log(P_{1}) + \beta_{12}\times log(P_{2})$
- $log(Q_2) = \beta_{20} + \beta_{21}\times log(P_{1}) + \beta_{22}\times log(P_{2})$

The price coefficients have two subscripts
- The first subscript refers to the equation
  - i.e. demand function for product $i$
- The second refers to the specific product/competitor

Pricing and Demand for Multiple Products

$log(Q_1) = \beta_{10} + \beta_{11}\times log(P_{1}) + \beta_{12}\times log(P_{2})$
$log(Q_2) = \beta_{20} + \beta_{21}\times log(P_{1}) + \beta_{22}\times log(P_{2})$

This means that:

$\beta_{10}$ is the $log$ of unit sales of Product 1 when $log(P_1) == 0$ & $log(P_2) == 0$
$\beta_{20}$ is the $log$ of unit sales of Product 2 when $log(P_1) == 0$ & $log(P_2) == 0$
$\beta_{11}$ is the…?
- Own price elasticity of Product 1
- $\beta_{21}$ is the own price elasticity of Product 2
- How much the sales of a product change when its own price changes
$\beta_{12}$ is the…?
- Cross price elasticity of Product 2 on Product 1
- How much the sales of a product change when its competitor’s price changes

Pricing and Demand for Multiple Products

The coefficients in these equations are price elasticities, just as in the simple multiplicative demand model discussed before:

$\beta_{ik} = \frac{\%\triangle Q_i}{\%\triangle P_k}$

Own and cross price elasticities

Own price elasticities are typically negative
- When the price of something increases, all else equal, sales fall
- This is why we expected $\beta_1$ to be negative in the single product case
Cross-price elasticities can be positive or negative
- When/why?
- Positive when products are substitutes for one another
  - (e.g. Coke vs. Pepsi)
  - Price of Coke increases, people switch to Pepsi
- Negative when products are compliments for one another
  - (e.g. razors and blades)

Implications of large cross-price elasticities

When price elasticity is large and positive:

Competitor’s product with high cross price elasticity?
- Competitive price change will have a large impact on the demand for our product
- Price competition will be tough — Low equilibrium prices
Product in our own product line with high price elasticity?
- Cannibalization of sales from other products in our product line
- Need to manage the prices of all products in product line simultaneously

Implications of large cross-price elasticities

When price elasticity is large and negative:

Competitor’s product with low cross price elasticity (< -2)?
- Price change will have a large impact on the demand for our product
- Tough spot to be in–can’t collude on prices
  - e.g., video games and consoles
Product in our own product line with low cross price elasticity (< -2)?
- Probably parts of a set
  - e.g., battery and charger
- Need to manage the prices of all products in product line simultaneously
- Potentially sell in bundles?

Implications of small cross-price elasticities

Product with low cross price elasticity?
- Price change will have a small impact on the demand for our product
- Products may not be true substitutes
  - e.g., Kashi cereal and Cookie Crisp
- Price competition will not be tough — Higher equilibrium prices
How can we get to this point?
- Advertising–make our product seem different
  - e.g., Gasoline, cell phone carriers, insurance providers, etc.
- If both products are ours, we could create optimally different products
  - e.g., combine with a conjoint analysis

Predicting changes in demand

To predict change in demand, we can use the change in prices:

$log(Q_1) = \beta_{10} + \beta_{11}\times log(P_{1}) + \beta_{12}\times log(P_{2})$
Came from: $Q_1 = A^{\beta_{10}} P_1^{\beta_{11}}P_2^{\beta_{12}}$
This was gross, but we can use it again:
- $\frac{Q'_1}{Q_1} = \frac{A^{\beta_{10}}P{'}_{12}^{\beta_{11}}P{'}_2^{\beta_{12}}}{A^{\beta_{10}}P_1^{\beta_{11}}P_2^{\beta_{12}}}$
- $P{'}_{i} = P_i\times (1 + \triangle P_i)$
- $\frac{Q'_1}{Q_1} = \frac{A^{\beta_{10}}P_1\times (1 + \triangle P_1)^{\beta_{11}}P_2\times (1 + \triangle P_2)^{\beta_{12}}}{A^{\beta_{10}}P_1^{\beta_{11}}P_2^{\beta_{12}}}$
- $\frac{Q'_1}{Q_1} = \frac{P_1\times (1 + \triangle P_1)^{\beta_{11}}P_2\times (1 + \triangle P_2)^{\beta_{12}}}{P_1^{\beta_{11}}P_2^{\beta_{12}}}$
- $\frac{Q'_1}{Q_1} = (1 + \triangle P_i)^{\beta_{11}}(1 + \triangle P_i)^{\beta_{12}}$

Predicting changes in demand

To predict change in demand, we can use the change in prices:

$\frac{Q'_1}{Q_1} = (1 + \triangle P_i)^{\beta_{11}}(1 + \triangle P_i)^{\beta_{12}}$

Or, in R:

change_in_Q1 <- (1 + change_in_P1) ^ beta11 * (1 + change_in_P2) ^ beta12

Workshop

We’ve got data about three cereals:

Frosted Flakes
Froot Loops
Kashi GoLean Crunch

Read in the data:

data <- read.csv("cereals_demand.csv")

Estimate

Demand for Froot Loops: Single-Product Case

$log(Q_{fl}) = \beta_{0} + \beta_{1}\times log(P_{fl})$

fl_single <- lm( data = data, log(sales_fl) ~ log(price_fl))

summary(fl_single)$coef |> round(3)

              Estimate Std. Error t value Pr(>|t|)
(Intercept)      9.006      0.534  16.858        0
log(price_fl)   -1.134      0.270  -4.195        0

Conclusions

Demand for Froot Loops: Single-Product Case

What is the own-price elasticity?
- -1.134
- 1% increase in price reduces sales by 1.134%
- Don’t increase prices!! Reduce them!
Look at the $R^2$ from this model. What is it?
- ~14%
- We explain 14% of the variation in sales with Froot Loop prices alone
- Can we do better?

Estimate

Demand for Froot Loops: Multiple-Product Case

$log(Q_{fl}) = \beta_{fl0} + \beta_{fl1}\times log(P_{fl}) + \beta_{fl2}\times log(P_{ff}) + \beta_{fl3}\times log(P_{k})$

fl_multiple <- lm( data = data, log(sales_fl) ~ log(price_fl) + log(price_ff) + log(price_k))

summary(fl_multiple)$coef |> round(3)

              Estimate Std. Error t value Pr(>|t|)
(Intercept)      6.323      0.867   7.291    0.000
log(price_fl)   -1.605      0.262  -6.124    0.000
log(price_ff)    1.255      0.223   5.637    0.000
log(price_k)     0.566      0.279   2.029    0.045

Conclusions

Demand for Froot Loops: Multiple-Product Case

What is the own-price elasticity?
- -1.6
- 1% increase in price reduces sales by 1.6%
- Don’t increase prices!! Reduce them!

Conclusions

Demand for Froot Loops: Multiple-Product Case

What is cross-price elasticity?
- Frosted Flakes:
  - 1.3
  - Frosted flakes prices impact us a lot!
  - Their price increases, our sales do too
- Kashi:
  - .6
  - Kashi prices don’t impact a lot
  - Their price increases, our sales do too
  - Not as competitive though

Conclusions

Demand for Froot Loops: Multiple-Product Case

What is $R^2$?
- .362
- We explain 14% of the variation in sales with all prices

With those around you:

Repeat this for Frosted flakes and Kashi

Estimate single and multiple product demand models
Note differences in elasticities
And $R^2$

Estimate

Demand for Frosted Flakes: Single-Product Case

ff_single <- lm( data = data, log(sales_ff) ~ log(price_ff))

summary(ff_single)$coef |> round(3)

              Estimate Std. Error t value Pr(>|t|)
(Intercept)      8.782      0.665  13.213    0.000
log(price_ff)   -1.239      0.345  -3.590    0.001

Conclusions

Demand for Frosted Flakes: Single-Product Case

What is the own-price elasticity?
- -1.24
- 1% increase in price reduces sales by 1.24%
- Don’t increase prices!! Reduce them!
Look at the $R^2$ from this model. What is it?
- ~10%

Estimate

Demand for Frosted Flakes: Multiple-Product Case

ff_multiple <- lm( data = data, log(sales_ff) ~ log(price_fl) + log(price_ff) + log(price_k))

summary(ff_multiple)$coef |> round(3)

              Estimate Std. Error t value Pr(>|t|)
(Intercept)      6.248      1.417   4.411    0.000
log(price_fl)    1.326      0.428   3.098    0.003
log(price_ff)   -1.674      0.364  -4.603    0.000
log(price_k)     0.355      0.456   0.779    0.438

Conclusions

Demand for Frosted Flakes: Multiple-Product Case

What is the own-price elasticity?
- -1.3
- 1% increase in price reduces sales by 1.3%
- Don’t increase prices!! Reduce them!

Conclusions

Demand for Frosted Flakes: Multiple-Product Case

What is cross-price elasticity?
- Kashi:
  - .35
  - Prices don’t impact us a lot
- Froot loops:
  - 1.67
  - Prices impact a lot

Conclusions

Demand for Frosted Flakes: Multiple-Product Case

What is $R^2$?
- .19
- We explain 19% of the variation in sales with all prices

Estimate

Demand for Kashi: Single-Product Case

k_single <- lm( data = data, log(sales_k) ~ log(price_k))

summary(k_single)$coef |> round(3)

             Estimate Std. Error t value Pr(>|t|)
(Intercept)     7.126      0.377  18.914        0
log(price_k)   -1.502      0.178  -8.446        0

Conclusions

Demand for Kashi: Single-Product Case

What is the own-price elasticity?
- -1.5
- 1% increase in price reduces sales by 1.5%
- Don’t increase prices!! Reduce them!
Look at the $R^2$ from this model. What is it?
- ~40%

Estimate

Demand for Kashi: Multiple-Product Case

k_multiple <- lm( data = data, log(sales_k) ~ log(price_fl) + log(price_ff) + log(price_k))

summary(k_multiple)$coef |> round(3)

              Estimate Std. Error t value Pr(>|t|)
(Intercept)      5.704      0.539  10.579    0.000
log(price_fl)    0.336      0.163   2.060    0.042
log(price_ff)    0.265      0.138   1.917    0.058
log(price_k)    -1.384      0.173  -7.980    0.000

Conclusions

Demand for Kashi: Multiple-Product Case

What is the own-price elasticity?
- -1.4
- 1% increase in price reduces sales by 1.4%
- Don’t increase prices!! Reduce them!

Conclusions

Demand for Kashi: Multiple-Product Case

What is cross-price elasticity?
- Frosted Flakes:
  - .27
  - Frosted flakes prices don’t impact us a lot
- Froot loops:
  - .34
  - Froot loops prices don’t impact a lot

Conclusions

Demand for Kashi: Multiple-Product Case

What is $R^2$?
- .47
- We explain 47% of the variation in sales with all prices

Change in prices

Now assume that:

Froot Loops increases its price by 5%
Kashi reduces its price by 15%

Calculate the percentage change in demand for each of the three products

Change in prices

For Froot Loops:

Define price changes

change_in_fl <- 0.05
change_in_ff <- 0
change_in_k <- -0.15

Change in prices

For Froot Loops:

Define price changes

change_in_fl <- 0.05
change_in_ff <- 0
change_in_k <- -0.15

Define demand changes

beta_fl_fl <- fl_multiple$coefficients[[2]]
beta_fl_ff <- fl_multiple$coefficients[[3]]
beta_fl_k <- fl_multiple$coefficients[[4]]

Change in prices

For Froot Loops:

Calculate ratio of sales

fl_ratio <- (1 + change_in_fl) ^ beta_fl_fl * (1 + change_in_ff) ^ beta_fl_ff * (1 + change_in_k) ^ beta_fl_k
fl_ratio

[1] 0.8433951

fl_pct <- 100*(fl_ratio-1)
fl_pct

[1] -15.66049

Repeat this for the other two products

Frosted Flakes

beta_ff_fl <- ff_multiple$coefficients[[2]]
beta_ff_ff <- ff_multiple$coefficients[[3]]
beta_ff_k <- ff_multiple$coefficients[[4]]

ff_ratio <- (1 + change_in_fl) ^ beta_ff_fl * (1 + change_in_ff) ^ beta_ff_ff * (1 + change_in_k) ^ beta_ff_k
ff_ratio

[1] 1.00707

ff_pct <- 100*(ff_ratio-1)
ff_pct

[1] 0.7069734

Kashi

beta_k_fl <- k_multiple$coefficients[[2]]
beta_k_ff <- k_multiple$coefficients[[3]]
beta_k_k <- k_multiple$coefficients[[4]]

k_ratio <- (1 + change_in_fl) ^ beta_k_fl * (1 + change_in_ff) ^ beta_k_ff * (1 + change_in_k) ^ beta_k_k
k_ratio

[1] 1.27296

k_pct <- 100*(k_ratio-1)
k_pct

[1] 27.29603

Estimated Profit

Assume that the base prices of three products are:

FL = $2.99
FF = $3.25
K = $4.99

And base sales levels are:

FL = 566
FF = 540
K = 302

Assuming the same percentage price changes, what do we expect i) demand and ii) profits for each product at the new prices to be?

Estimated Profit

Define baseline prices and demands

price_FL <- 2.99
price_FF <- 3.25
price_K <- 4.99

demand_FL <- 566
demand_FF <- 540
demand_K <- 302

Estimated Profit

Define new prices and demands

price_FL_2 <- price_FL * (1 + change_in_fl)
price_FF_2 <- price_FF * (1 + change_in_ff)
price_K_2 <- price_K * (1 + change_in_k)

demand_FL_2 <- demand_FL * fl_ratio
demand_FF_2 <- demand_FF * ff_ratio
demand_K_2 <- demand_K * k_ratio

demand_FL_2; demand_FF_2; demand_K_2

[1] 477.3616

[1] 543.8177

[1] 384.434

Estimated Profit

Estimate profit

Using marginal costs of:
- $1.50 per box of Frosted Flakes and Froot Loops
- $2.00 per box of Kashi

mcost_ff <- 1.5
mcost_fl <- 1.5
mcost_k <- 2

pft_ff_2 <- demand_FF_2*(price_FF_2-mcost_ff); pft_ff_2

[1] 951.6809

pft_fl_2 <- demand_FL_2*(price_FL_2-mcost_fl); pft_fl_2

[1] 782.6344

pft_k_2 <- demand_K_2*(price_K_2-mcost_k); pft_k_2

[1] 861.7088

Estimated Profit

Compare to baseline:

pft_ff_1 <- demand_FF*(price_FF-mcost_ff); pft_ff_1

[1] 945

pft_fl_1 <- demand_FL*(price_FL-mcost_fl); pft_fl_1

[1] 843.34

pft_k_1 <- demand_K*(price_K-mcost_k); pft_k_1

[1] 902.98

Conclusions

We can use demand models to predict demand for single products, as well as multiple.
Adding parameters for other products allows us to estimate demand in a competitive landscape.
We use these models (rather than conjoint) when we have observational data about sales and product attributes
- Price is one attribute. We could have done this with others.

Conclusions

Elasticities > 0 indicate as attribute increases, so do sales
Elasticities < 0 indicate as attribute increases, sales decrease
Elasticities < -1 indicate that we should decrease prices, as sales will increase more to make up
We could have extended this example a lot, to simulate profits, entire markets, etc.